Harry is the Boy Who Lived
Prior to the series, Voldemort was basically taking over magical Britain. He was a terror, with very few people (the Order of the Phoenix, basically) opposing him.
Then Voldemort found James, Lily, and baby Harry Potter. James was killed, Lily was killed, and Harry... wasn't. Instead, Voledmort disappeared. The little baby had survived and apparently defeated the Dark Lord with nothing more than a lightning-shaped scar.
This alone was enough to make Harry a living legend in the magical community. The majority of this information is revealed in the first book/movie, Harry Potter and the Sorcerer's Stone.
Sybill Trelawney's First Prophecy
In Harry Potter and the Order of the Phoenix is it revealed that when Sybill Trelawney (the divinations professor) was being interviewed for her job at Hogwarts, she made a prophecy:
The one with the power to vanquish the Dark Lord approaches... born to those who have thrice defied him, born as the seventh month dies... and the Dark Lord will mark him as his equal, but he will have power the Dark Lord knows not... and either must die at the hand of the other for neither can live while the other survives... the one with the power to vanquish the Dark Lord will be born as the seventh month dies...
In the movies, the prophecy is similar, but slightly different:
The one with the power to vanquish the Dark Lord approaches... and the Dark Lord shall mark him as his equal but he shall have power the Dark Lord knows not... for neither can live while the other survives...
(The movie version of the prophecy doesn't mention being born in July, but Dubledore later claims that it is also part of the prophecy.)
Combined with the fact that Harry is the "Boy Who Lived", all of the characters who are aware of the prophecy understand that it refers to Harry, making him even more important: the universe says he's important, after all!
The prophecy could have also been referring to Neville Longbottom, who was born within days of Harry and whose parents had also defied Voldemort on three occasions. However, after hearing the prophecy himself Voldemort chose to attack the Potters because Harry was a half-blood, just like Voldemort was (Neville was pure-blood). By making that decision, Voldemort marked Harry as his equal, completing the prophecy's description of the child.
So, not only did Harry free Britain from tyrannical rule by Voldemort, the universe claimed that Harry was the key to defeating Voldemort. Thus, he was a Very Important Wizard™, regardless of what skills he had or didn't have.